Publication Cover
Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 7
208
Views
0
CrossRef citations to date
0
Altmetric
Research Article

A non-Newtonian Noether's symmetry theorem

ORCID Icon
Pages 1934-1941 | Received 02 Aug 2021, Accepted 19 Nov 2021, Published online: 06 Dec 2021

References

  • Grossman M, Katz R. Non-Newtonian calculus. Pigeon Cove (MA): Lee Press; 1972.
  • Bashirov AE, Mısırlı E, Tandoğdu Y, et al. On modeling with multiplicative differential equations. Appl Math J Chinese Univ Ser B. 2011;26(4):425–438.
  • Czachor M. Unifying aspects of generalized calculus. Entropy. 2020;22(10):1180.
  • Mora M, Córdova-Lepe F, Del-Valle R. A non-Newtonian gradient for contour detection in images with multiplicative noise. Pattern Recognit Lett. 2012;33(10):1245–1256.
  • Ozyapici A, Bilgehan B. Finite product representation via multiplicative calculus and its applications to exponential signal processing. Numer Algorithms. 2016;71(2):475–489.
  • Pinto M, Torres R, Campillay-Llanos W, et al. Applications of proportional calculus and a non-Newtonian logistic growth model. Proyecciones. 2020;39(6):1471–1513.
  • Campillay-Llanos W, Guevara F, Pinto M, et al. Differential and integral proportional calculus: how to find a primitive for f(x)=1/2πe−(1/2)x2. Int J Math Educ Sci Technol. 2021;52(3):463–476.
  • Waseem M, Noor MA, Shah FA, et al. An efficient technique to solve nonlinear equations using multiplicative calculus. Turkish J Math. 2018;42(2):679–691.
  • Grossman M. Bigeometric calculus. Rockport (MA): Archimedes Foundation; 1983.
  • Torres DFM. On a non-Newtonian calculus of variations. Axioms. 2021;10(3):1–15.
  • Pap E. Generalized real analysis and its applications. Int J Approx Reason. 2008;47(3):368–386.
  • Tekin S, Başar F. Certain sequence spaces over the non-Newtonian complex field. Abstr Appl Anal. 2013;11:Article ID 739319.
  • Duyar C, SağIr B, Oğur O. Some basic topological properties on non-Newtonian real line. Br J Math Comput Sci. 2015;9(4):300–307.
  • Binbaşıoǧlu D, Demiriz S, Türkoǧlu D. Fixed points of non-Newtonian contraction mappings on non-Newtonian metric spaces. J Fixed Point Theory Appl. 2016;18(1):213–224.
  • Güngör N. Some geometric properties of the non-Newtonian sequence spaces lp(N). Math Slovaca. 2020;70(3):689–696.
  • Sağır B, Erdoğan F. On non-Newtonian power series and its applications. Konuralp J Math. 2020;8(2):294–303.
  • Burgin M, Czachor M. Non-diophantine arithmetics in mathematics, physics and psychology. Singapore: World Scientific; 2021.
  • Noether E. Invariante variations probleme. Nachr Ges Wiss Göttingen Math-Phys Kl. 1918;1918:235–257.
  • Noether E. Invariant variation problems. Trans Theory Stat Phys. 1971;1(3):186–207.
  • Alekseev VM, Tikhomirov VM, Fomin SV. Optimal control. New York (NY): Consultants Bureau; 1987. (Contemporary Soviet mathematics).
  • Gelfand IM, Fomin SV. Calculus of variations. Englewood Cliffs (NJ): Prentice-Hall Inc.; 1963.
  • Jost J, Li-Jost X. Calculus of variations. Cambridge: Cambridge University Press; 1998. (Cambridge studies in advanced mathematics; vol. 64).
  • Sarlet W, Cantrijn F. Generalizations of Noether's theorem in classical mechanics. SIAM Rev. 1981;23(4):467–494.
  • Santos SPS, Martins N, Torres DFM. Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem. Discrete Contin Dyn Syst. 2015;35(9):4593–4610.
  • Torres DFM. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Commun Pure Appl Anal. 2004;3(3):491–500.
  • Torres DFM. Gauge symmetries and Noether currents in optimal control. Appl Math E-Notes. 2003;3:49–57.
  • Torres DFM. Conservation laws in optimal control. In: Dynamics, bifurcations, and control (Kloster Irsee, 2001). Berlin: Springer; 2002. p. 287–296. (Lecture notes in control and information sciences; vol. 273).
  • Torres DFM. On the Noether theorem for optimal control. Eur J Control. 2002;8(1):56–63.
  • Torres DFM. Quasi-invariant optimal control problems. Port Math (NS). 2004;61(1):97–114.
  • Kastrup HA. The contributions of Emmy Noether, Felix Klein and Sophus Lie to the modern concept of symmetries in physical systems. In: Symmetries in physics (1600–1980) (San Feliu de Guíxols, 1983). Barcelona: Univ. Autònoma Barcelona; 1987. p. 113–163.
  • Gouveia PDF, Torres DFM. Automatic computation of conservation laws in the calculus of variations and optimal control. Comput Methods Appl Math. 2005;5(4):387–409.
  • Gouveia PDF, Torres DFM, Rocha EAM. Symbolic computation of variational symmetries in optimal control. Control Cybernet. 2006;35(4):831–849.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.